In this paper we apply proof mining techniques to compute, in the setting of CAT
$(\unicode[STIX]{x1D705})$
spaces (with
$\unicode[STIX]{x1D705}>0$
), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.

[32]TaoT.. Soft analysis, hard analysis, and the finite convergence principle. Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence, RI, 2008, Essay first posted May 23, 2007.